issubrg2 - Metamath Proof Explorer

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Theorem issubrg2 20044

Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)

**Hypotheses**
Ref	Expression
issubrg2.b	⊢ 𝐵 = (Base‘𝑅)
issubrg2.o	⊢ 1 = (1_r‘𝑅)
issubrg2.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
issubrg2	⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝑅,𝑦 𝑥, · ,𝑦 Allowed substitution hints: 𝐵(𝑥,𝑦) 1 (𝑥,𝑦)

Proof of Theorem issubrg2 Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgsubg 20030	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrg2.o	. . . 4 ⊢ 1 = (1_r‘𝑅)
3	2	subrg1cl 20032	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
4		issubrg2.t	. . . . . 6 ⊢ · = (._r‘𝑅)
5	4	subrgmcl 20036	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
6	5	3expb 1119	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
7	6	ralrimivva 3123	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
8	1, 3, 7	3jca 1127	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9		simpl 483	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10		simpr1 1193	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11		eqid 2738	. . . . . . 7 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
12	11	subgbas 18759	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾_s 𝐴)))
13	10, 12	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾_s 𝐴)))
14		eqid 2738	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
15	11, 14	ressplusg 17000	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+_g‘𝑅) = (+_g‘(𝑅 ↾_s 𝐴)))
16	10, 15	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+_g‘𝑅) = (+_g‘(𝑅 ↾_s 𝐴)))
17	11, 4	ressmulr 17017	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → · = (._r‘(𝑅 ↾_s 𝐴)))
18	10, 17	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (._r‘(𝑅 ↾_s 𝐴)))
19	11	subggrp 18758	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Grp)
20	10, 19	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾_s 𝐴) ∈ Grp)
21		simpr3 1195	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
23	22	eleq1d 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24		oveq2 7283	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
25	24	eleq1d 2823	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
26	23, 25	rspc2v 3570	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
27	21, 26	syl5com 31	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28	27	3impib 1115	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29		issubrg2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
30	29	subgss 18756	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
31	10, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
32	31	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
33	31	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
34	31	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
35	32, 33, 34	3anim123d 1442	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
36	35	imp 407	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
37	29, 4	ringass 19803	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
38	37	adantlr 712	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
39	36, 38	syldan 591	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
40	29, 14, 4	ringdi 19805	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
41	40	adantlr 712	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
42	36, 41	syldan 591	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
43	29, 14, 4	ringdir 19806	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
44	43	adantlr 712	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
45	36, 44	syldan 591	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
46		simpr2 1194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1 ∈ 𝐴)
47	32	imp 407	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐵)
48	29, 4, 2	ringlidm 19810	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
49	48	adantlr 712	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
50	47, 49	syldan 591	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → ( 1 · 𝑢) = 𝑢)
51	29, 4, 2	ringridm 19811	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
52	51	adantlr 712	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
53	47, 52	syldan 591	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑢 · 1 ) = 𝑢)
54	13, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53	isringd 19824	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾_s 𝐴) ∈ Ring)
55	31, 46	jca 512	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))
56	29, 2	issubrg 20024	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
57	9, 54, 55, 56	syl21anbrc 1343	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
58	57	ex 413	. 2 ⊢ (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
59	8, 58	impbid2 225	1 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3887 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 ↾_s cress 16941 +_gcplusg 16962 ._rcmulr 16963 Grpcgrp 18577 SubGrpcsubg 18749 1_rcur 19737 Ringcrg 19783 SubRingcsubrg 20020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-subg 18752 df-mgp 19721 df-ur 19738 df-ring 19785 df-subrg 20022

This theorem is referenced by: opprsubrg 20045 subrgint 20046 issubrg3 20052 issubrngd2 20459 cnsubrglem 20648 mplsubrg 21211 mplind 21278 dmatsrng 21650 scmatsrng 21669 scmatsrng1 21672 cpmatsrgpmat 21870

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